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The Finite Difference Method for Boundary Value Problem with Singularity

  • Viktor A. RukavishnikovEmail author
  • Elena I. Rukavishnikova
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9045)

Abstract

For boundary value problems with singularity, we developed the theory of finite difference schemes based on concept of an \(R_\nu \)-generalized solution. The difference scheme is constructed, the rate of convergence of the approximate solution to the \(R_\nu \)-generalized solution in the norm of the Sobolev weighted space is established.

Keywords

The \(R_\nu \)-generalized solution for BVP Finite difference method 

References

  1. 1.
    Rukavishnikov, V.A.: On a weighted estimate of the rate of convergence of difference schemes. Sov. Math. Dokl. 22, 826–829 (1986)Google Scholar
  2. 2.
    Rukavishnikov, V.A.: On differentiability properties of an \({R}_\nu \)-generalized solution of Dirichlet problem. Sov. Math. Dokl. 40, 653–655 (1990)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Rukavishnikov, V.A.: On a weighted estimates of error of difference schemes for Helmholtz’s equation. Numer. Anal. Math. Model. 24, 397–408 (1990). Banach Center Publications, WarsawMathSciNetGoogle Scholar
  4. 4.
    Rukavishnikov, V.A.: Study of difference schemes for Dirichlet problem Sobolev’s weight spaces. Sibiring J. Comput. Math. 1, 191–204 (1992)zbMATHGoogle Scholar
  5. 5.
    Rukavishnikov, V.A.: On the Dirichlet problem for the second order elliptic equation with noncoordinated degeneration of input data. Differ. Equ. 32, 406–412 (1996)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Rukavishnikov, V.A.: On the uniqueness of \({R}_\nu \)-generalized solution for boundary value problem with non-coordinated degeneration of the input data. Dokl. Math. 63, 68–70 (2001)Google Scholar
  7. 7.
    Rukavishnikov, V.A., Ereklintsev, A.G.: On the coercivity of the \({R}_\nu \)-generalized solution of the first boundary value problem with coordinated degeneration of the input data. Differ. Equ. 41, 1757–1767 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Rukavishnikov, V.A., Kuznetsova, E.V.: Coercive estimate for a boundary value problem with noncoordinated degeneration of the data. Differ. Equ. 43, 550–560 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Rukavishnikov, V.A., Kuznetsova, E.V.: The \({R}_\nu \)-generalized solution of a boundary value problem with a singularity belongs to the space \({W}^{k+2}_{2,\nu +\beta /2+k+1}(\Omega,\delta )\). Differ. Equ. 45, 913–917 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Rukavishnikov, V.A., Rukavishnikova, E.I.: The finite element method for the first boundary value problem with compatible degeneracy of the input data. Russ. Acad. Sci. Dokl. Math. 50, 335–339 (1995)MathSciNetGoogle Scholar
  11. 11.
    Rukavishnikov, V.A., Kuznetsova, E.V.: A finite element method scheme for boundary value problems with noncoordinated degeneration of input data. Numer. Anal. Appl. 2, 250–259 (2009)CrossRefGoogle Scholar
  12. 12.
    Rukavishnikov, V.A., Rukavishnikova, H.I.: The finite element method for a boundary value problem with strong singularity. J. Comp. Appl. Math. 234, 2870–2882 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Rukavishnikov, V.A.: On differential properties \({R}_{\nu }\)-generalized solution of the Dirichlet problem with coordinated degeneration of the input data. ISRN Math. Anal. 243724, 18 (2011). doi: 10.5402/2011/243724 MathSciNetGoogle Scholar
  14. 14.
    Rukavishnikov, V.A., Mosolapov, A.O.: New numerical method for solving time-harmonic Maxwell equations with strong singularity. J. Comput. Phys. 231, 2438–2448 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Rukavishnikov, V.A., Mosolapov, A.O.: Weighted edge finite element method for Maxwell’s equations with strong singularity. Dokl. Math. 87, 156–159 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Rukavishnikov, V.A., Nikolaev, S.G.: Weighted finite element method for an elasticity problem with singularity. Dokl. Math. 88, 705–709 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Rukavishnikov, V.A., Rukavishnikova, H.I.: On the error estimation of the finite element method for the boundary value problems with singularity in the lebesgue weighted space. Numer. Funct. Anal. Optim. 34, 1328–1347 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Rukavishnikov, V.A.: On the existence and uniqueness of \(R_\nu \)-generalized solution for boundary value problem with the non-coordinated degeneration of the initial data. Dokl. Akad. Nauk. 458(3), 261–263 (2014)MathSciNetGoogle Scholar
  19. 19.
    Bramble, J.H., Hilbert, S.R.: Estimation of linear functional on Sobolev spaces with applications to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7, 112–124 (1970)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Viktor A. Rukavishnikov
    • 1
    Email author
  • Elena I. Rukavishnikova
    • 1
  1. 1.Computing Center, Far-Eastern BranchRussian Academy of SciencesKhabarovskRussian Federation

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